calculus 2 series and sequences practice test

(answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. endobj 979.2 979.2 979.2 272 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 When you have completed the free practice test, click 'View Results' to see your results. Strip out the first 3 terms from the series \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{2^{ - n}}}}{{{n^2} + 1}}} \). 531.3 531.3 531.3] We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. 441.3 461.2 353.6 557.3 473.4 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272] L7s[AQmT*Z;HK%H0yqt1r8 We will illustrate how partial sums are used to determine if an infinite series converges or diverges. Which of the following sequences is NOT a geometric sequence? . Given that n=0 1 n3 +1 = 1.6865 n = 0 1 n 3 + 1 = 1.6865 determine the value of n=2 1 n3 +1 . Donate or volunteer today! bmkraft7. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. 326.4 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 (answer), Ex 11.2.9 Compute \(\sum_{n=1}^\infty {3^n\over 5^{n+1}}\). Which of the following is the 14th term of the sequence below? To log in and use all the features of Khan Academy, please enable JavaScript in your browser. ZrNRG{I~(iw%0W5b)8*^ yyCCy~Cg{C&BPsTxp%p Strategy for Series In this section we give a general set of guidelines for determining which test to use in determining if an infinite series will converge or diverge. >> /Widths[606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 652.8 598 757.6 622.8 552.8 . 489.6 272 489.6 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 /Type/Font 500 388.9 388.9 277.8 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 Infinite sequences and series | AP/College Calculus BC - Khan Academy Chapters include Linear 272 761.6 462.4 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 722.2 777.8 777.8 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). We will also see how we can use the first few terms of a power series to approximate a function. >> Khan Academy is a 501(c)(3) nonprofit organization. Solving My Calc 2 Exam#3 (Sequence, Infinite Series & Power Series) Math 106 (Calculus II): old exams. ]^e-V!2 F. 6.5E: Exercises for Comparison Test - Mathematics LibreTexts Math 1242: Calculus II - University of North Carolina at Charlotte /Length 465 S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< /Length 1722 /BaseFont/BPHBTR+CMMI12 (answer), Ex 11.9.4 Find a power series representation for \( 1/(1-x)^3\). endstream endobj 208 0 obj <. Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. 12 0 obj Note as well that there really isnt one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines. More on Sequences In this section we will continue examining sequences. How many bricks are in the 12th row? 18 0 obj PDF Practice Problems Series & Sequences - MR. SOLIS' WEEBLY /Subtype/Type1 /FirstChar 0 1) \(\displaystyle \sum^_{n=1}a_n\) where \(a_n=\dfrac{2}{n . endobj %%EOF Calculus II - Series - The Basics (Practice Problems) - Lamar University /Filter /FlateDecode 750 750 750 1044.4 1044.4 791.7 791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Solution. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. 5.3.3 Estimate the value of a series by finding bounds on its remainder term. >> 489.6 489.6 272 272 761.6 489.6 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 /FontDescriptor 20 0 R Defining convergent and divergent infinite series, Determining absolute or conditional convergence, Finding Taylor polynomial approximations of functions, Radius and interval of convergence of power series, Finding Taylor or Maclaurin series for a function. Series The Basics In this section we will formally define an infinite series. 489.6 272 489.6 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 Taylor Series In this section we will discuss how to find the Taylor/Maclaurin Series for a function. Let the factor without dx equal u and the factor with dx equal dv. >> /Length 569 Strategies for Testing Series - University of Texas at Austin Ex 11.8.1 \(\sum_{n=0}^\infty n x^n\) (answer), Ex 11.8.2 \(\sum_{n=0}^\infty {x^n\over n! A ball is dropped from an unknown height (h) and it repeatedly bounces on the floor. 413.2 531.3 826.4 295.1 354.2 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 /Name/F1 1277.8 555.6 1000 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). Calculus II For Dummies Cheat Sheet - dummies SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . Each term is the sum of the previous two terms. 326.4 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 /Widths[777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). /FontDescriptor 14 0 R 26 0 obj << /FontDescriptor 11 0 R Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). 508.8 453.8 482.6 468.9 563.7 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 Each term is the difference of the previous two terms. xYKs6W(MCG:9iIO=(lkFRI$x$AMN/" J?~i~d cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. 70 terms. Which of the sequences below has the recursive rule where each number is the previous number times 2? Derivatives, Integrals, Sequences & Series, and Vector Valued Functions. Here is a list of all the sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section. 531.3 590.3 472.2 590.3 472.2 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 !A1axw)}p]WgxmkFftu We will also give many of the basic facts and properties well need as we work with sequences. /LastChar 127 /Widths[611.8 816 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 707.2 571.2 544 544 The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. If youd like a pdf document containing the solutions the download tab above contains links to pdfs containing the solutions for the full book, chapter and section. copyright 2003-2023 Study.com. Question 5 5. >> 590.3 767.4 795.8 795.8 1091 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 /BaseFont/UNJAYZ+CMR12 Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. hb```9B 7N0$K3 }M[&=cx`c$Y&a YG&lwG=YZ}w{l;r9P"J,Zr]Ngc E4OY%8-|\C\lVn@`^) E 3iL`h`` !f s9B`)qLa0$FQLN$"H&8001a2e*9y,Xs~z1111)QSEJU^|2n[\\5ww0EHauC8Gt%Y>2@ " Worksheets. /FirstChar 0 531.3 590.3 560.8 414.1 419.1 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 Ex 11.1.3 Determine whether {n + 47 n} . Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. /Widths[611.8 816 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 707.2 571.2 544 544 Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges. We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). /BaseFont/CQGOFL+CMSY10 /Name/F3 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. Calculus II - Sequences and Series Flashcards | Quizlet in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. Published by Wiley. \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. (answer), Ex 11.11.3 Find the first three nonzero terms in the Taylor series for \(\tan x\) on \([-\pi/4,\pi/4]\), and compute the guaranteed error term as given by Taylor's theorem. /FirstChar 0 272 816 544 489.6 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 Power Series In this section we will give the definition of the power series as well as the definition of the radius of convergence and interval of convergence for a power series. Sequences & Series in Calculus Chapter Exam. (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. )^2\over n^n}\) (answer). ,vEmO8/OuNVRaLPqB.*l. Each term is the product of the two previous terms. endobj stream 805.6 805.6 1277.8 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 %PDF-1.5 A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. At this time, I do not offer pdf's for solutions to individual problems. Our mission is to provide a free, world-class education to anyone, anywhere. For problems 1 - 3 perform an index shift so that the series starts at n = 3 n = 3. 272 761.6 462.4 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 /FontDescriptor 23 0 R 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. Example 1. Course summary; . Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). Determine whether the following series converge or diverge. Boundary Value Problems & Fourier Series, 8.3 Periodic Functions & Orthogonal Functions, 9.6 Heat Equation with Non-Zero Temperature Boundaries, 1.14 Absolute Value Equations and Inequalities, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). /LastChar 127 Good luck! 4 avwo/MpLv) _C>5p*)i=^m7eE. Level up on all the skills in this unit and collect up to 2000 Mastery points! >> 252 0 obj <>stream (answer). If you're seeing this message, it means we're having trouble loading external resources on our website. n = 1 n 2 + 2 n n 3 + 3 n . endstream (answer), Ex 11.3.12 Find an \(N\) so that \(\sum_{n=2}^\infty {1\over n(\ln n)^2}\) is between \(\sum_{n=2}^N {1\over n(\ln n)^2}\) and \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). 68 0 obj % /Name/F6 Determine whether the sequence converges or diverges. 826.4 531.3 958.7 1076.8 826.4 295.1 295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. (answer). At this time, I do not offer pdfs for solutions to individual problems. Infinite series are sums of an infinite number of terms. Given that \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865\) determine the value of \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \). Khan Academy is a 501(c)(3) nonprofit organization. (answer). If you . (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). Math 106 (Calculus II): old exams | Mathematics | Bates College << /FontDescriptor 8 0 R >> sCA%HGEH[ Ah)lzv<7'9&9X}xbgY[ xI9i,c_%tz5RUam\\6(ke9}Yv`B7yYdWrJ{KZVUYMwlbN_>[wle\seUy24P,PyX[+l\c $w^rvo]cYc@bAlfi6);;wOF&G_. /Length 1247 (answer). All rights reserved. n = 1 n2 + 2n n3 + 3n2 + 1. 816 816 272 299.2 489.6 489.6 489.6 489.6 489.6 792.7 435.2 489.6 707.2 761.6 489.6 556.5 425.2 527.8 579.5 613.4 636.6 609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 /Length 200 Part II. (answer), Ex 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. Images. 5.3.1 Use the divergence test to determine whether a series converges or diverges. Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. Donate or volunteer today! Determine whether the series is convergent or divergent. Which of the following sequences follows this formula. About this unit. Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. (answer), Ex 11.1.6 Determine whether \(\left\{{2^n\over n! 272 816 544 489.6 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 >> PDF Schaums Outline Of Differential Equations 4th Edition Schaums Outline Comparison tests. 31 terms. (answer). You appear to be on a device with a "narrow" screen width (, 2.4 Equations With More Than One Variable, 2.9 Equations Reducible to Quadratic in Form, 4.1 Lines, Circles and Piecewise Functions, 1.5 Trig Equations with Calculators, Part I, 1.6 Trig Equations with Calculators, Part II, 3.6 Derivatives of Exponential and Logarithm Functions, 3.7 Derivatives of Inverse Trig Functions, 4.10 L'Hospital's Rule and Indeterminate Forms, 5.3 Substitution Rule for Indefinite Integrals, 5.8 Substitution Rule for Definite Integrals, 6.3 Volumes of Solids of Revolution / Method of Rings, 6.4 Volumes of Solids of Revolution/Method of Cylinders, A.2 Proof of Various Derivative Properties, A.4 Proofs of Derivative Applications Facts, 7.9 Comparison Test for Improper Integrals, 9. /Subtype/Type1 62 0 obj endobj Which rule represents the nth term in the sequence 9, 16, 23, 30? 590.3 885.4 885.4 295.1 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 If it converges, compute the limit. For each of the following series, determine which convergence test is the best to use and explain why. 11.E: Sequences and Series (Exercises) - Mathematics LibreTexts /Type/Font Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. We will also determine a sequence is bounded below, bounded above and/or bounded. endobj With an outline format that facilitates quick and easy review, Schaum's Outline of Calculus, Seventh Edition helps you understand basic concepts and get the extra practice you need to excel in these courses. A proof of the Ratio Test is also given. stream Series | Calculus 2 | Math | Khan Academy 9 0 obj Power Series and Functions In this section we discuss how the formula for a convergent Geometric Series can be used to represent some functions as power series. /FontDescriptor 17 0 R The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. 2 6 points 2. /LastChar 127 >> Some infinite series converge to a finite value. Math 129 - Calculus II Worksheets - University of Arizona When you have completed the free practice test, click 'View Results' to see your results. Differentiate u to find du, and integrate dv to find v. Use the formula: Evaluate the right side of this equation to solve the integral. /Filter /FlateDecode x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J PDF Ap Calculus Ab Bc Kelley Copy - gny.salvationarmy.org endstream 5.3 The Divergence and Integral Tests - Calculus Volume 2 - OpenStax Section 10.3 : Series - Basics. UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm 15 0 obj 762 689.7 1200.9 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 Series are sums of multiple terms. Choose your answer to the question and click 'Continue' to see how you did. Calculus II-Sequences and Series. stream Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. Then click 'Next Question' to answer the next question. Study Online AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.2 -The Integral Test and p-Series Study Notes Prepared by AP Teachers Skip to content . A proof of the Integral Test is also given. (answer). 45 0 obj Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. Each review chapter is packed with equations, formulas, and examples with solutions, so you can study smarter and score a 5! Series are sums of multiple terms. << 24 0 obj Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . /LastChar 127 Ex 11.11.4 Show that \(\cos x\) is equal to its Taylor series for all \(x\) by showing that the limit of the error term is zero as N approaches infinity. copyright 2003-2023 Study.com. Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. /FirstChar 0 << Mediansandsuch - Medians - MATH 126 Medians and Such Let X be - Studocu Sequences and Series. PDF Read Free Answers To Algebra 2 Practice B Ellipses (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). Harmonic series and p-series. Then click 'Next Question' to answer the next question. Our mission is to provide a free, world-class education to anyone, anywhere. Note that some sections will have more problems than others and some will have more or less of a variety of problems. endobj If it converges, compute the limit. Ex 11.1.2 Use the squeeze theorem to show that limn n! It turns out the answer is no. Good luck! Which one of these sequences is a finite sequence? If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Good luck! endobj Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . Then click 'Next Question' to answer the next question. Calculus II - Series & Sequences (Practice Problems) - Lamar University Your instructor might use some of these in class. For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. PDF Calculus II Series - Things to Consider - California State University In the previous section, we determined the convergence or divergence of several series by . Good luck! We will also briefly discuss how to determine if an infinite series will converge or diverge (a more in depth discussion of this topic will occur in the next section). (You may want to use Sage or a similar aid.) MULTIPLE CHOICE: Circle the best answer. Math C185: Calculus II (Tran) 6: Sequences and Series 6.5: Comparison Tests 6.5E: Exercises for Comparison Test Expand/collapse global location 6.5E: Exercises for Comparison Test . To log in and use all the features of Khan Academy, please enable JavaScript in your browser. /Subtype/Type1 5.3.2 Use the integral test to determine the convergence of a series. A review of all series tests. Then determine if the series converges or diverges. AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.4 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 Choose your answer to the question and click 'Continue' to see how you did. The following is a list of worksheets and other materials related to Math 129 at the UA. To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. (answer), Ex 11.11.1 Find a polynomial approximation for \(\cos x\) on \([0,\pi]\), accurate to \( \pm 10^{-3}\) (answer), Ex 11.11.2 How many terms of the series for \(\ln x\) centered at 1 are required so that the guaranteed error on \([1/2,3/2]\) is at most \( 10^{-3}\)? We will also give many of the basic facts, properties and ways we can use to manipulate a series. Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? Then we can say that the series diverges without having to do any extra work. Determine whether the series converge or diverge. 8 0 obj Parametric Equations and Polar Coordinates, 9.5 Surface Area with Parametric Equations, 9.11 Arc Length and Surface Area Revisited, 10.7 Comparison Test/Limit Comparison Test, 12.8 Tangent, Normal and Binormal Vectors, 13.3 Interpretations of Partial Derivatives, 14.1 Tangent Planes and Linear Approximations, 14.2 Gradient Vector, Tangent Planes and Normal Lines, 15.3 Double Integrals over General Regions, 15.4 Double Integrals in Polar Coordinates, 15.6 Triple Integrals in Cylindrical Coordinates, 15.7 Triple Integrals in Spherical Coordinates, 16.5 Fundamental Theorem for Line Integrals, 3.8 Nonhomogeneous Differential Equations, 4.5 Solving IVP's with Laplace Transforms, 7.2 Linear Homogeneous Differential Equations, 8. 979.2 489.6 489.6 489.6] Don't all infinite series grow to infinity? 722.6 693.1 833.5 795.8 382.6 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 Integral test. 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] Ex 11.4.1 \(\sum_{n=1}^\infty {(-1)^{n-1}\over 2n+5}\) (answer), Ex 11.4.2 \(\sum_{n=4}^\infty {(-1)^{n-1}\over \sqrt{n-3}}\) (answer), Ex 11.4.3 \(\sum_{n=1}^\infty (-1)^{n-1}{n\over 3n-2}\) (answer), Ex 11.4.4 \(\sum_{n=1}^\infty (-1)^{n-1}{\ln n\over n}\) (answer), Ex 11.4.5 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^3}\) to two decimal places. Estimating the Value of a Series In this section we will discuss how the Integral Test, Comparison Test, Alternating Series Test and the Ratio Test can, on occasion, be used to estimating the value of an infinite series. Learning Objectives. The Ratio Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. Divergence Test. 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. 531.3 531.3 531.3 295.1 295.1 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. stream 888.9 888.9 888.9 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 << We also discuss differentiation and integration of power series. Alternating Series Test For series of the form P ( 1)nb n, where b n is a positive and eventually decreasing sequence, then X ( 1)nb n converges ()limb n = 0 POWER SERIES De nitions X1 n=0 c nx n OR X1 n=0 c n(x a) n Radius of convergence: The radius is de ned as the number R such that the power series . endobj 1111.1 472.2 555.6 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 If it con-verges, nd the limit. xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f BS!.eSloaY&h&Urm!U3L%g@'>`|$ogJ endstream endobj startxref nn = 0. This will work for a much wider variety of function than the method discussed in the previous section at the expense of some often unpleasant work. %PDF-1.5 For problems 1 3 perform an index shift so that the series starts at \(n = 3\). To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. %PDF-1.2 We will also give the Divergence Test for series in this section. (answer), Ex 11.3.11 Find an \(N\) so that \(\sum_{n=1}^\infty {\ln n\over n^2}\) is between \(\sum_{n=1}^N {\ln n\over n^2}\) and \(\sum_{n=1}^N {\ln n\over n^2} + 0.005\). 1. /Subtype/Type1 |: The Ratio Test shows us that regardless of the choice of x, the series converges. endobj Ex 11.7.3 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\). &/ r %PDF-1.5 % 555.6 577.8 577.8 597.2 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 }\) (answer), Ex 11.8.3 \(\sum_{n=1}^\infty {n!\over n^n}x^n\) (answer), Ex 11.8.4 \(\sum_{n=1}^\infty {n!\over n^n}(x-2)^n\) (answer), Ex 11.8.5 \(\sum_{n=1}^\infty {(n!

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